Existence of Positive Solutions to Semilinear Equations with Sublinear Nonlinearities and Compact Positivity-Improving Resolvent

We prove a Brézis–Oswald type existence theorem for positive solutions of semilinear equations in an abstract setting in which the underlying linear operator has a compact positivity-improving resolvent. The assumptions imposed on the sublinear nonlinearity are comparable to those used in the classical elliptic theory, but no additional regularizing properties of the resolvent, such as ultracontractivity or smoothing, are required. The proof combines the method of sub- and supersolutions with spectral ideas of Brézis–Oswald type and several new arguments adapted to the abstract operator-theoretic framework. In this way, the paper provides a bridge between order-theoretic methods and the classical energy-based theory of sublinear problems.